Trigonometry

Learn, then practise

The original worked lessons remain the main explanations. Use these focused reminders to check the conditions before you start.

Exact values, quadrants and principal inverse values

Use reference angles and quadrant signs for exact ratios.
Interpret negative angles and angles beyond one revolution.
Distinguish principal inverse values from a complete solution set.

Use the quadrant to choose signs. A unit-circle point is \((\cos x,\sin x)\). Principal ranges: \(\sin^{-1}\) uses \(\displaystyle [- \frac{\pi }{2}, \frac{\pi }{2}]\), \(\cos^{-1}\) uses \([0,\pi]\), and \(\tan^{-1}\) uses \(\displaystyle (- \frac{\pi }{2}, \frac{\pi }{2})\). Inverse notation does not mean reciprocal.

Read the full notes and 3 worked examples · Next: short practice

Trigonometric graphs, periods and transformations

Read amplitude, midline, period and full coordinates.
Respect a restricted domain when finding a range.
Sketch tangent branches separately and identify asymptotes.

For \(y=a+b\sin(cx+d)\), amplitude is \(|b|\) and period is \(\displaystyle \frac{2\pi }{|c|}\). Tangent has period \(\displaystyle \frac{\pi }{|c|}\), no amplitude, and is undefined when its cosine denominator is zero. A restricted interval may not attain the full sine or cosine range.

Read the full notes and 3 worked examples · Next: short practice

Identities, factorisation and valid domains

Prove identities using the two P1 core identities.
Keep restrictions from original denominators.
Factor equations without losing zero-factor solutions.

Use \(\sin^2x+\cos^2x=1\) and \(\displaystyle \tan x= \frac{\sin x}{\cos x}\). Start an identity proof from one side. Cancellation does not fill a hole in the original domain. Check a zero factor before dividing by it.

Read the full notes and 3 worked examples · Next: short practice

Trigonometric equations on a stated interval

Find every solution in a finite interval.
Transform both interval endpoints for a multiple or shifted argument.
Handle degrees, radians, principal values and open endpoints consistently.

Work on the full argument interval before converting back to \(x\). Use every relevant sine/cosine branch and the tangent period. Check included endpoints, rejected endpoints and the original expression. A single inverse-calculator value is rarely the complete answer.

Read the full notes and 3 worked examples · Next: short practice

Trigonometric equations that become quadratics

Use one trigonometric ratio as an algebraic variable.
Reject impossible sine and cosine values and retain both signs when appropriate.
Count distinct angle solutions, including endpoint and repeated-root cases.

Solve the algebraic equation in \(u=\sin x\), \(\cos x\) or \(\tan x\). Sine and cosine must lie in \([-1,1]\); tangent is not subject to that bound. Each admissible ratio still needs every angle in the stated interval. Repeated algebraic roots do not create additional distinct angles.

Read the full notes and 3 worked examples · Next: short practice